This is a problem from Hartshorne's Algebraic Geometry Chapter III Ex.6.4:

Let $X$ be a noetherian scheme, and suppose that every coherent sheaf on $X$ is a quotient of a locally free sheaf. Then for any $\mathcal{G} \in \mathcal{Mod}(X)$, show that the $\delta$-functor $\mathscr{Ext}^{i}(\cdot,\mathcal{G})$ from $\mathcal{Coh}(X)$ to $\mathcal{Mod}(X)$, is a contravariant universal $\delta$-functor.

My attempt and failure:

By the hint from this book, we want to show that this $\delta$-functor is coeffacable. Now by condition we have a surjection from a locally free sheaf $\mathcal{F'}$ to $\mathcal{F}$. From Hartshorne's book we know that locally free sheaf of finite rank has vanished higher Ext sheaves, so I want to find a locally free sheaf of finite rank of $\mathcal{F'}$. By extension of coherent sheaf we can find a coherent subsheaf of $\mathcal{F'}$. But it seems that this sheaf not necessary becomes locally free again, and neither has a vanished higher Ext sheave from a priori.

Any help is appreciated.

Edit: I've thought about that Hartshorne implies the locally free sheaf to have finite rank since he also refers it as "$\mathcal{Coh}(X)$ has enough locally frees". If this is the case I appreciate a counter example of the original statement above.


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